# The common ratio of a G.P. is 3 and the last term is 486.

Question:

The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.

Solution:

Here, common ratio, = 3

nth term, an = 486

Sn = 728

$a_{n}=486$

$\Rightarrow a r^{n-1}=486$

$\Rightarrow a(3)^{n-1}=486$

$\Rightarrow a(3)^{n}=486 \times 3$

$\Rightarrow a(3)^{n}=1458$    ....(i)

Now, $S_{n}=728$

$\Rightarrow 728=a\left(\frac{3^{n}-1}{3-1}\right)$

$\Rightarrow 728=\left\{\frac{a(3)^{n}-a}{2}\right\}$

$\Rightarrow 1456=a(3)^{n-1}-a$

$\Rightarrow 1456=1458-a$        [From (i)]

$\Rightarrow a=1458-1456$

$\Rightarrow a=2$

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